logging in or signing up Orbital Mechanics Overview 2 Ubert Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINTLite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 1446 Category: Education License: All Rights Reserved Like it (1) Dislike it (0) Added: January 11, 2008 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... By: faiyyaj (25 month(s) ago) please send me this presentation on faiyyaj11@gmail.com i need it Saving..... Post Reply Close Saving..... Edit Comment Close By: anup009 (33 month(s) ago) please sent me ppt. of Orbital Mechanics Overview 2 in my id anuprocks_n@yahoo.co.in soon. Saving..... Post Reply Close Saving..... Edit Comment Close By: ahmed9901 (39 month(s) ago) Hi, Can you please send me the Power Point for the Bicycle work that you have done. at the following address ahmed9901@yahoo.com I will be grateful. Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript Orbital Mechanics Overview 2: Orbital Mechanics Overview 2 MAE 155B G. NacouziOrbital Mechanics Overview 2: Orbital Mechanics Overview 2 Summary of first quarter overview Keplerian motion Classical orbit parameters Orbital perturbations Central body observation Coverage examples using Excel Project workshop Introduction: Orbital Mechanics: Introduction: Orbital Mechanics Motion of satellite is influenced by the gravity field of multiple bodies, however, two body assumption is usually sufficient. Earth orbiting satellite Two Body approach: Central body is earth, assume it has only gravitational influence on S/C, assume M >> m (M, m ~ mass of earth & S/C) Gravity effects of secondary bodies including sun, moon and other planets in solar system are ignored Gravitational potential function is given by: = GM/r Solution assumes bodies are spherically symmetric, point sources (Earth oblateness not accounted for) Only gravity and centrifugal forces are present Two Body Motion (or Keplerian Motion): Two Body Motion (or Keplerian Motion) Closed form solution for 2 body exists, no explicit soltn exists for N >2, numerical approach needed Gravitational field on body is given by: Fg = M m G/R2 where, M~ Mass of central body; m~ Mass of Satellite G~ Universal gravity constant R~ distance between centers of bodies For a S/C in Low Earth Orbit (LEO), the gravity forces are: Earth: 0.9 g Sun: 6E-4 g Moon: 3E-6 g Jupiter: 3E-8 g Elliptical Orbit Geometry & Nomenclature: Elliptical Orbit Geometry & Nomenclature Periapsis Apoapsis Line of Apsides R a c V Rp b Line of Apsides connects Apoapsis, central body & Periapsis Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature) S/C position defined by R & , is called true anomaly R = [Rp (1+e)]/[1+ e cos()]Elliptical Orbit Definition: Elliptical Orbit Definition Orbit is defined using the 6 classical orbital elements: Eccentricity, semi-major axis, true anomaly: position of SC on the orbit inclination, i, is the angle between orbit plane and equatorial plane Argument of Periapsis (). Angle from Ascending Node (AN) to Periapsis. AN: Pt where S/C crosses equatorial plane South to North - Longitude of Ascending Node ()~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending nodeSources of Orbital Perturbations: Sources of Orbital Perturbations Several external forces cause perturbation to spacecraft orbit 3rd body effects, e.g., sun, moon, other planets Unsymmetrical central bodies (‘oblateness’ caused by rotation rate of body): Earth: Requator = 6378 km, Rpolar = 6357 km Space Environment: Solar Pressure, drag from rarefied atmosphere Reference: C. Brown, ‘Elements of SC Design’ Relative Importance of Orbit Perturbations: Relative Importance of Orbit Perturbations J2 term accounts for effect from oblate earth Principal effect above 100 km altitude Other terms may also be important depending on application, mission, etc... Reference: Spacecraft Systems Engineering, Fortescue & StarkPrincipal Orbital Perturbations: Principal Orbital Perturbations Earth ‘oblateness’ results in an unsymmetric gravity potential given by: where ae = equatorial radius, Pn ~ Legendre Polynomial Jn ~ zonal harmonics, w ~ sin (SC declination) J2 term causes measurable perturbation which must be accounted for. Main effects: Regression of nodes Rotation of apsides Note: J2~1E-3, J3~1E-6Orbital Perturbation Effects: Regression of Nodes: Orbital Perturbation Effects: Regression of Nodes Regression of Nodes: Equatorial bulge causes component of gravity vector acting on SC to be slightly out of orbit plane This out of orbit plane component causes a slight precession of the orbit plane. The resulting orbital rotation is called regression of nodes and is approximated using the dominant gravity harmonics term, J2Regression of Nodes: Regression of Nodes Regression of nodes is approximated by: Where, ~ Longitude of the ascending node; R~ Mean equatorial radius J2 ~ Zonal coeff.(for earth = 0.001082) n ~ mean motion (sqrt(GM/a3)), a~ semimajor axis Note: Although regression rate is small for Geo., it is cumulative and must be accounted forOrbital Perturbation: Rotation of Apsides: Orbital Perturbation: Rotation of Apsides Rotation of apsides caused by earth oblateness is similar to regression of nodes. The phenomenon is caused by a higher acceleration near the equator and a resulting overshoot at periapsis. This only occurs in elliptical orbits. The rate of rotation is given by: Ground Track: Ground Track Defined as the trace of nadir positions, as a function of time, on the central body. Ground track is influenced by: S/C orbit Rotation of central body Orbit perturbations Trace is calculated using spherical trigonometry (no perturbances) sin (La) = sin (i) sin ALa Lo = + asin(tan (La)/tan(i))+Re where: Ala ~ (ascending node to SC) ~ Longitude of ascending node I ~ Inclination Re~Earth rotation rate= 0.0042t (add to west. longitudes, subtract for eastern longitude)Example Ground Trace: Example Ground TraceSpacecraft Horizon: Spacecraft Horizon SC horizon forms a circle on the spherical surface of the central body, within circle: SC can be seen from central body Line of sight communication can be established SC can observe the central bodyCentral Body Observation: Central Body Observation From simple trigonometry: sin(h) = Rs/(Rs+hs) Dh = (Rs+hs) cos(h) Sw~ Swath width = 2 h Rs You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
Orbital Mechanics Overview 2 Ubert Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINTLite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 1446 Category: Education License: All Rights Reserved Like it (1) Dislike it (0) Added: January 11, 2008 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... By: faiyyaj (25 month(s) ago) please send me this presentation on faiyyaj11@gmail.com i need it Saving..... Post Reply Close Saving..... Edit Comment Close By: anup009 (33 month(s) ago) please sent me ppt. of Orbital Mechanics Overview 2 in my id anuprocks_n@yahoo.co.in soon. Saving..... Post Reply Close Saving..... Edit Comment Close By: ahmed9901 (39 month(s) ago) Hi, Can you please send me the Power Point for the Bicycle work that you have done. at the following address ahmed9901@yahoo.com I will be grateful. Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript Orbital Mechanics Overview 2: Orbital Mechanics Overview 2 MAE 155B G. NacouziOrbital Mechanics Overview 2: Orbital Mechanics Overview 2 Summary of first quarter overview Keplerian motion Classical orbit parameters Orbital perturbations Central body observation Coverage examples using Excel Project workshop Introduction: Orbital Mechanics: Introduction: Orbital Mechanics Motion of satellite is influenced by the gravity field of multiple bodies, however, two body assumption is usually sufficient. Earth orbiting satellite Two Body approach: Central body is earth, assume it has only gravitational influence on S/C, assume M >> m (M, m ~ mass of earth & S/C) Gravity effects of secondary bodies including sun, moon and other planets in solar system are ignored Gravitational potential function is given by: = GM/r Solution assumes bodies are spherically symmetric, point sources (Earth oblateness not accounted for) Only gravity and centrifugal forces are present Two Body Motion (or Keplerian Motion): Two Body Motion (or Keplerian Motion) Closed form solution for 2 body exists, no explicit soltn exists for N >2, numerical approach needed Gravitational field on body is given by: Fg = M m G/R2 where, M~ Mass of central body; m~ Mass of Satellite G~ Universal gravity constant R~ distance between centers of bodies For a S/C in Low Earth Orbit (LEO), the gravity forces are: Earth: 0.9 g Sun: 6E-4 g Moon: 3E-6 g Jupiter: 3E-8 g Elliptical Orbit Geometry & Nomenclature: Elliptical Orbit Geometry & Nomenclature Periapsis Apoapsis Line of Apsides R a c V Rp b Line of Apsides connects Apoapsis, central body & Periapsis Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature) S/C position defined by R & , is called true anomaly R = [Rp (1+e)]/[1+ e cos()]Elliptical Orbit Definition: Elliptical Orbit Definition Orbit is defined using the 6 classical orbital elements: Eccentricity, semi-major axis, true anomaly: position of SC on the orbit inclination, i, is the angle between orbit plane and equatorial plane Argument of Periapsis (). Angle from Ascending Node (AN) to Periapsis. AN: Pt where S/C crosses equatorial plane South to North - Longitude of Ascending Node ()~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending nodeSources of Orbital Perturbations: Sources of Orbital Perturbations Several external forces cause perturbation to spacecraft orbit 3rd body effects, e.g., sun, moon, other planets Unsymmetrical central bodies (‘oblateness’ caused by rotation rate of body): Earth: Requator = 6378 km, Rpolar = 6357 km Space Environment: Solar Pressure, drag from rarefied atmosphere Reference: C. Brown, ‘Elements of SC Design’ Relative Importance of Orbit Perturbations: Relative Importance of Orbit Perturbations J2 term accounts for effect from oblate earth Principal effect above 100 km altitude Other terms may also be important depending on application, mission, etc... Reference: Spacecraft Systems Engineering, Fortescue & StarkPrincipal Orbital Perturbations: Principal Orbital Perturbations Earth ‘oblateness’ results in an unsymmetric gravity potential given by: where ae = equatorial radius, Pn ~ Legendre Polynomial Jn ~ zonal harmonics, w ~ sin (SC declination) J2 term causes measurable perturbation which must be accounted for. Main effects: Regression of nodes Rotation of apsides Note: J2~1E-3, J3~1E-6Orbital Perturbation Effects: Regression of Nodes: Orbital Perturbation Effects: Regression of Nodes Regression of Nodes: Equatorial bulge causes component of gravity vector acting on SC to be slightly out of orbit plane This out of orbit plane component causes a slight precession of the orbit plane. The resulting orbital rotation is called regression of nodes and is approximated using the dominant gravity harmonics term, J2Regression of Nodes: Regression of Nodes Regression of nodes is approximated by: Where, ~ Longitude of the ascending node; R~ Mean equatorial radius J2 ~ Zonal coeff.(for earth = 0.001082) n ~ mean motion (sqrt(GM/a3)), a~ semimajor axis Note: Although regression rate is small for Geo., it is cumulative and must be accounted forOrbital Perturbation: Rotation of Apsides: Orbital Perturbation: Rotation of Apsides Rotation of apsides caused by earth oblateness is similar to regression of nodes. The phenomenon is caused by a higher acceleration near the equator and a resulting overshoot at periapsis. This only occurs in elliptical orbits. The rate of rotation is given by: Ground Track: Ground Track Defined as the trace of nadir positions, as a function of time, on the central body. Ground track is influenced by: S/C orbit Rotation of central body Orbit perturbations Trace is calculated using spherical trigonometry (no perturbances) sin (La) = sin (i) sin ALa Lo = + asin(tan (La)/tan(i))+Re where: Ala ~ (ascending node to SC) ~ Longitude of ascending node I ~ Inclination Re~Earth rotation rate= 0.0042t (add to west. longitudes, subtract for eastern longitude)Example Ground Trace: Example Ground TraceSpacecraft Horizon: Spacecraft Horizon SC horizon forms a circle on the spherical surface of the central body, within circle: SC can be seen from central body Line of sight communication can be established SC can observe the central bodyCentral Body Observation: Central Body Observation From simple trigonometry: sin(h) = Rs/(Rs+hs) Dh = (Rs+hs) cos(h) Sw~ Swath width = 2 h Rs